Two sides and an angle are given below. Determine whether the given information results in one triangles, or no triangle at all. Solve any resulting triangle(s).&nbsp;b=4,&nbsp;c=5,&nbsp;B=30∘&nbsp;Choose the appropriate option below and, if needed, fill in the blanks to complete your selection.A) A single triangle is produced, where&nbsp;C≈?∘,&nbsp;A≈?∘, and&nbsp;a≈?&nbsp;B) Two triangles are produced, where the triangle with the smaller angle C has&nbsp;C1≈?∘,&nbsp;A1≈?∘, and&nbsp;a1≈?, and the triangle with the larger angle C has&nbsp;C2≈?∘,&nbsp;A2≈?∘,&nbsp;and&nbsp;a2≈?&nbsp;C) No triangles are produced.

Question

Two sides and an angle are given below. Determine whether the given information results in one triangles, or no triangle at all. Solve any resulting triangle(s).&amp;nbsp;b=4,&amp;nbsp;c=5,&amp;nbsp;B=30∘&amp;nbsp;Choose the appropriate option below and, if needed, fill in the blanks to complete your selection.A) A single triangle is produced, where&amp;nbsp;C≈?∘,&amp;nbsp;A≈?∘, and&amp;nbsp;a≈?&amp;nbsp;B) Two triangles are produced, where the triangle with the smaller angle C has&amp;nbsp;C1≈?∘,&amp;nbsp;A1≈?∘, and&amp;nbsp;a1≈?, and the triangle with the larger angle C has&amp;nbsp;C2≈?∘,&amp;nbsp;A2≈?∘,&amp;nbsp;and&amp;nbsp;a2≈?&amp;nbsp;C) No triangles are produced.

William Yazzie · Accepted Answer

Step 1  This problem can be solved using Law of sines of triangle. The Law of Sines is the relationship between the sides and angles of non-right triangles.  Simply, it states that the ratio of the length of a side of a triangle to the sine of the angle opposite that side is the same for all sides and angles in a given triangle.    for the triangle △ABC shown in figure with side lengths a, b, c the law of sines is given as,  asin⁡A=bsin⁡B=csin⁡C  Step 2  In the question, side b=4, c=5, B=30∘  applying law of sines,  4sin⁡30∘=5sin⁡C  ⇒40.5=5sin⁡C  ⇒8=5sin⁡C  ⇒sin⁡C=58=0.625  ⇒C=sin−1(0.625)=38.68∘  The sum of angles of triangle =180∘  therefore  ∠A=180−38.68−30  ∠A=11.32∘  now using Law of sines we can find length of third side.  asin⁡A=bsin⁡B  ⇒asin⁡111.32=4sin⁡30  ⇒a0.931=40.5  ⇒a=40.5×0.931=7.45  Step 3  Now let us see if we can have another triangle. We have B=30∘ given  Now subtract already calculated C from 180∘  ⇒C2=180∘−30∘−141.32∘  therefore the third angle will be,  A2=180∘−30∘−141.32∘  A2=8.68∘  a2sin⁡A=bsin⁡B  ⇒a2sin⁡8.68=4sin⁡30  ⇒a20.15=40.5  ⇒a2=40.5×0.15=1.207  Step 4  Therefore there are two triangles possible for the given measurements b=4, c=5, B=30∘  Triangle 1  C1=38.68∘  A1=111.32∘

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